Calculation of electromagnetic field components for a fundamental Gaussian beam

Using a perturbative method, we carefully calculate seventh- and ninth-order expressions for electromagnetic field components for a fundamental Gaussian beam (i.e., a focused TEM00 mode laser beam) propagating in the medium analogous to vacuum. The interaction of a single electron and a focused laser pulse in vacuum has a promising application in the design of a powerful electron accelerator which could compete with the traditional ones. For this, ninth-order accuracy in the expansion describing the focused laser beam is required. When the vector potential corresponds to polarization along the direction of propagation, the number of electromagnetic field components can be reduced from 5 or 6 to 3. Furthermore, we find two rules obeyed by all orders of the vector potential, which greatly simplify the calculation procedure of the vector potential order by order, and make it possible to investigate the behavior of high orders.